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Properties

Label 10T18
Degree $10$
Order $200$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^2 : C_8$

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Show commands: Magma

magma: G := TransitiveGroup(10, 18);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $18$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5^2 : C_8$
CHM label:  $[5^{2}:4]2_{2}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,7,9,3)(2,4,8,6), (1,4,3,2,9,6,7,8)(5,10), (2,4,6,8,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$8$:  $C_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

10T18 x 2, 20T56 x 3, 25T20, 40T171 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 4, 4, 1, 1 $ $25$ $4$ $( 3, 5, 9, 7)( 4, 6,10, 8)$
$ 4, 4, 1, 1 $ $25$ $4$ $( 3, 7, 9, 5)( 4, 8,10, 6)$
$ 2, 2, 2, 2, 1, 1 $ $25$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)$
$ 5, 1, 1, 1, 1, 1 $ $8$ $5$ $( 2, 4, 6, 8,10)$
$ 8, 2 $ $25$ $8$ $( 1, 2)( 3, 4, 7, 8, 9,10, 5, 6)$
$ 8, 2 $ $25$ $8$ $( 1, 2)( 3, 6, 5,10, 9, 8, 7, 4)$
$ 8, 2 $ $25$ $8$ $( 1, 2)( 3, 8, 5, 4, 9, 6, 7,10)$
$ 8, 2 $ $25$ $8$ $( 1, 2)( 3,10, 7, 6, 9, 4, 5, 8)$
$ 5, 5 $ $8$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 5, 5 $ $8$ $5$ $( 1, 3, 5, 7, 9)( 2, 8, 4,10, 6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $200=2^{3} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  200.40
magma: IdentifyGroup(G);
 
Character table:    
      2  3  3  3  3  .   3   3   3   3  .  .
      5  2  .  .  .  2   .   .   .   .  2  2

        1a 4a 4b 2a 5a  8a  8b  8c  8d 5b 5c
     2P 1a 2a 2a 1a 5a  4b  4a  4a  4b 5b 5c
     3P 1a 4b 4a 2a 5a  8c  8d  8a  8b 5b 5c
     5P 1a 4a 4b 2a 1a  8d  8c  8b  8a 1a 1a
     7P 1a 4b 4a 2a 5a  8b  8a  8d  8c 5b 5c

X.1      1  1  1  1  1   1   1   1   1  1  1
X.2      1  1  1  1  1  -1  -1  -1  -1  1  1
X.3      1 -1 -1  1  1   A  -A  -A   A  1  1
X.4      1 -1 -1  1  1  -A   A   A  -A  1  1
X.5      1  A -A -1  1   B  /B -/B  -B  1  1
X.6      1  A -A -1  1  -B -/B  /B   B  1  1
X.7      1 -A  A -1  1 -/B  -B   B  /B  1  1
X.8      1 -A  A -1  1  /B   B  -B -/B  1  1
X.9      8  .  .  .  3   .   .   .   . -2 -2
X.10     8  .  .  . -2   .   .   .   . -2  3
X.11     8  .  .  . -2   .   .   .   .  3 -2

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)

magma: CharacterTable(G);